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The small $p$-adic Simpson correspondence in the semi-stable reduction case

Mao Sheng, Yupeng Wang

Abstract

We generalize several known results on small Simpson correspondence for smooth formal schemes over $\calO_C$ to the case for semi-stable formal schemes. More precisely, for a liftable semi-stable formal scheme $\frakX$ over $\calO_C$ with generic fiber $X$, we establish (1) an equivalence between the category of Hitchin-small integral $v$-bundles on $X_{v}$ and the category of Hitchin-small Higgs bundles on $\frakX_{\et}$, generalizing the previous work of Min--Wang, and (2) an equivalence between the moduli stack of $v$-bundles on $X_{v}$ and the moduli stack of rational Higgs bundles on $\frakX_{\et}$ (equivalently, moduli stack of Higgs bundles on $X_{\et}$), generalizing the previous work of Anschütz--Heuer--Le Bras.

The small $p$-adic Simpson correspondence in the semi-stable reduction case

Abstract

We generalize several known results on small Simpson correspondence for smooth formal schemes over to the case for semi-stable formal schemes. More precisely, for a liftable semi-stable formal scheme over with generic fiber , we establish (1) an equivalence between the category of Hitchin-small integral -bundles on and the category of Hitchin-small Higgs bundles on , generalizing the previous work of Min--Wang, and (2) an equivalence between the moduli stack of -bundles on and the moduli stack of rational Higgs bundles on (equivalently, moduli stack of Higgs bundles on ), generalizing the previous work of Anschütz--Heuer--Le Bras.

Paper Structure

This paper contains 23 sections, 44 theorems, 297 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. An integral $p$-adic Simpson correspondence
  5. A stacky $p$-adic Simpson correspondence
  6. Organization
  7. Notations
  8. Basic set-up
  9. Canonical log structure on perfectoid affinoid algebras
  10. Semi-stable formal schemes over $A^+$
  11. Period sheaves
  12. Integral Faltings' extension
  13. Period sheaves
  14. A local Simpson correspondence
  15. $\Gamma$-representations over $R_S^+$ v.s. $\Gamma$-representations over $\widehat{R}_{\infty,S}^+$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let ${\mathfrak X}$ be a semi-stable formal scheme over ${\mathcal{O}}_C$ of relative dimension $d$. Suppose that it admits a flat lifting (as a formal log-scheme) $\widetilde{{\mathfrak X}}$ over ${\mathbf A}_{2,K}$. Then there exists a period sheaf $({\mathcal{O}}\widehat{{\mathbb C}}_{{\mathrm{pd between the category ${\mathrm L}{\mathrm S}^{\text{H-sm}}({\mathfrak X},{\widehat{{\mathcal{O}}}_X

Theorems & Definitions (99)

  • Theorem 1.1: Theorem \ref{['thm:integral Simpson']}
  • Remark 1.2
  • Theorem 1.3: Corollary \ref{['cor:bounded torsion']} and Theorem \ref{['thm:truncated comparison']}
  • Theorem 1.4: Theorem \ref{['thm:DI']}
  • Remark 1.5
  • Theorem 1.6: Theorem \ref{['thm:stacky Simpson']}
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Theorem \ref{['thm:rational Simpson']} and Lemma \ref{['lem:Hitchin-small vs twisted Hitchin-small']}
  • Definition 2.1
  • ...and 89 more